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Mirrors > Home > MPE Home > Th. List > axlowdim1 | Structured version Visualization version GIF version |
Description: The lower dimension axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 26673. (Contributed by Scott Fenton, 22-Apr-2013.) |
Ref | Expression |
---|---|
axlowdim1 | ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10629 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 1 | fconst6 6562 | . . 3 ⊢ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ |
3 | elee 26607 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {1}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ)) | |
4 | 2, 3 | mpbiri 259 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ∈ (𝔼‘𝑁)) |
5 | 0re 10631 | . . . 4 ⊢ 0 ∈ ℝ | |
6 | 5 | fconst6 6562 | . . 3 ⊢ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ |
7 | elee 26607 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {0}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ)) | |
8 | 6, 7 | mpbiri 259 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {0}) ∈ (𝔼‘𝑁)) |
9 | ax-1ne0 10594 | . . . . . . 7 ⊢ 1 ≠ 0 | |
10 | 9 | neii 3015 | . . . . . 6 ⊢ ¬ 1 = 0 |
11 | 1ex 10625 | . . . . . . 7 ⊢ 1 ∈ V | |
12 | 11 | sneqr 4763 | . . . . . 6 ⊢ ({1} = {0} → 1 = 0) |
13 | 10, 12 | mto 198 | . . . . 5 ⊢ ¬ {1} = {0} |
14 | elnnuz 12270 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
15 | eluzfz1 12902 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
16 | 14, 15 | sylbi 218 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁)) |
17 | 16 | ne0d 4298 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
18 | rnxp 6020 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {1}) = {1}) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {1}) = {1}) |
20 | rnxp 6020 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {0}) = {0}) | |
21 | 17, 20 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {0}) = {0}) |
22 | 19, 21 | eqeq12d 2834 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0}) ↔ {1} = {0})) |
23 | 13, 22 | mtbiri 328 | . . . 4 ⊢ (𝑁 ∈ ℕ → ¬ ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) |
24 | rneq 5799 | . . . 4 ⊢ (((1...𝑁) × {1}) = ((1...𝑁) × {0}) → ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) | |
25 | 23, 24 | nsyl 142 | . . 3 ⊢ (𝑁 ∈ ℕ → ¬ ((1...𝑁) × {1}) = ((1...𝑁) × {0})) |
26 | 25 | neqned 3020 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0})) |
27 | neeq1 3075 | . . 3 ⊢ (𝑥 = ((1...𝑁) × {1}) → (𝑥 ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ 𝑦)) | |
28 | neeq2 3076 | . . 3 ⊢ (𝑦 = ((1...𝑁) × {0}) → (((1...𝑁) × {1}) ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0}))) | |
29 | 27, 28 | rspc2ev 3632 | . 2 ⊢ ((((1...𝑁) × {1}) ∈ (𝔼‘𝑁) ∧ ((1...𝑁) × {0}) ∈ (𝔼‘𝑁) ∧ ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0})) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
30 | 4, 8, 26, 29 | syl3anc 1363 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 ∅c0 4288 {csn 4557 × cxp 5546 ran crn 5549 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 ℕcn 11626 ℤ≥cuz 12231 ...cfz 12880 𝔼cee 26601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-z 11970 df-uz 12232 df-fz 12881 df-ee 26604 |
This theorem is referenced by: btwndiff 33385 |
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